Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,2,Mod(13,72)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("72.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 72.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 61.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 72.61
Dual form 72.2.n.a.13.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.366025 + 1.36603i) q^{2} +(0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(1.73205 - 1.00000i) q^{5} +(2.36603 + 0.633975i) q^{6} +(-2.00000 + 3.46410i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(2.00000 + 2.00000i) q^{10} +(-2.59808 - 1.50000i) q^{11} +3.46410i q^{12} +(-1.73205 + 1.00000i) q^{13} +(-5.46410 - 1.46410i) q^{14} -3.46410i q^{15} +(2.00000 - 3.46410i) q^{16} +5.00000 q^{17} +(3.00000 - 3.00000i) q^{18} -1.00000i q^{19} +(-2.00000 + 3.46410i) q^{20} +(3.46410 + 6.00000i) q^{21} +(1.09808 - 4.09808i) q^{22} +(-1.00000 - 1.73205i) q^{23} +(-4.73205 + 1.26795i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-2.00000 - 2.00000i) q^{26} -5.19615 q^{27} -8.00000i q^{28} +(4.73205 - 1.26795i) q^{30} +(2.00000 + 3.46410i) q^{31} +(5.46410 + 1.46410i) q^{32} +(-4.50000 + 2.59808i) q^{33} +(1.83013 + 6.83013i) q^{34} +8.00000i q^{35} +(5.19615 + 3.00000i) q^{36} -2.00000i q^{37} +(1.36603 - 0.366025i) q^{38} +3.46410i q^{39} +(-5.46410 - 1.46410i) q^{40} +(2.50000 + 4.33013i) q^{41} +(-6.92820 + 6.92820i) q^{42} +(9.52628 + 5.50000i) q^{43} +6.00000 q^{44} +(-5.19615 - 3.00000i) q^{45} +(2.00000 - 2.00000i) q^{46} +(3.00000 - 5.19615i) q^{47} +(-3.46410 - 6.00000i) q^{48} +(-4.50000 - 7.79423i) q^{49} +(-1.36603 - 0.366025i) q^{50} +(4.33013 - 7.50000i) q^{51} +(2.00000 - 3.46410i) q^{52} +(-1.90192 - 7.09808i) q^{54} -6.00000 q^{55} +(10.9282 - 2.92820i) q^{56} +(-1.50000 - 0.866025i) q^{57} +(0.866025 - 0.500000i) q^{59} +(3.46410 + 6.00000i) q^{60} +(-10.3923 - 6.00000i) q^{61} +(-4.00000 + 4.00000i) q^{62} +12.0000 q^{63} +8.00000i q^{64} +(-2.00000 + 3.46410i) q^{65} +(-5.19615 - 5.19615i) q^{66} +(-2.59808 + 1.50000i) q^{67} +(-8.66025 + 5.00000i) q^{68} -3.46410 q^{69} +(-10.9282 + 2.92820i) q^{70} -6.00000 q^{71} +(-2.19615 + 8.19615i) q^{72} +9.00000 q^{73} +(2.73205 - 0.732051i) q^{74} +(0.866025 + 1.50000i) q^{75} +(1.00000 + 1.73205i) q^{76} +(10.3923 - 6.00000i) q^{77} +(-4.73205 + 1.26795i) q^{78} +(7.00000 - 12.1244i) q^{79} -8.00000i q^{80} +(-4.50000 + 7.79423i) q^{81} +(-5.00000 + 5.00000i) q^{82} +(-3.46410 - 2.00000i) q^{83} +(-12.0000 - 6.92820i) q^{84} +(8.66025 - 5.00000i) q^{85} +(-4.02628 + 15.0263i) q^{86} +(2.19615 + 8.19615i) q^{88} -14.0000 q^{89} +(2.19615 - 8.19615i) q^{90} -8.00000i q^{91} +(3.46410 + 2.00000i) q^{92} +6.92820 q^{93} +(8.19615 + 2.19615i) q^{94} +(-1.00000 - 1.73205i) q^{95} +(6.92820 - 6.92820i) q^{96} +(-0.500000 + 0.866025i) q^{97} +(9.00000 - 9.00000i) q^{98} +9.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 6 q^{6} - 8 q^{7} - 8 q^{8} - 6 q^{9} + 8 q^{10} - 8 q^{14} + 8 q^{16} + 20 q^{17} + 12 q^{18} - 8 q^{20} - 6 q^{22} - 4 q^{23} - 12 q^{24} - 2 q^{25} - 8 q^{26} + 12 q^{30} + 8 q^{31} + 8 q^{32}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.366025 + 1.36603i 0.258819 + 0.965926i
\(3\) 0.866025 − 1.50000i 0.500000 − 0.866025i
\(4\) −1.73205 + 1.00000i −0.866025 + 0.500000i
\(5\) 1.73205 − 1.00000i 0.774597 − 0.447214i −0.0599153 − 0.998203i \(-0.519083\pi\)
0.834512 + 0.550990i \(0.185750\pi\)
\(6\) 2.36603 + 0.633975i 0.965926 + 0.258819i
\(7\) −2.00000 + 3.46410i −0.755929 + 1.30931i 0.188982 + 0.981981i \(0.439481\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) −2.00000 − 2.00000i −0.707107 − 0.707107i
\(9\) −1.50000 − 2.59808i −0.500000 − 0.866025i
\(10\) 2.00000 + 2.00000i 0.632456 + 0.632456i
\(11\) −2.59808 − 1.50000i −0.783349 − 0.452267i 0.0542666 − 0.998526i \(-0.482718\pi\)
−0.837616 + 0.546259i \(0.816051\pi\)
\(12\) 3.46410i 1.00000i
\(13\) −1.73205 + 1.00000i −0.480384 + 0.277350i −0.720577 − 0.693375i \(-0.756123\pi\)
0.240192 + 0.970725i \(0.422790\pi\)
\(14\) −5.46410 − 1.46410i −1.46034 − 0.391298i
\(15\) − 3.46410i − 0.894427i
\(16\) 2.00000 − 3.46410i 0.500000 − 0.866025i
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 3.00000 − 3.00000i 0.707107 − 0.707107i
\(19\) − 1.00000i − 0.229416i −0.993399 − 0.114708i \(-0.963407\pi\)
0.993399 − 0.114708i \(-0.0365932\pi\)
\(20\) −2.00000 + 3.46410i −0.447214 + 0.774597i
\(21\) 3.46410 + 6.00000i 0.755929 + 1.30931i
\(22\) 1.09808 − 4.09808i 0.234111 − 0.873713i
\(23\) −1.00000 − 1.73205i −0.208514 − 0.361158i 0.742732 − 0.669588i \(-0.233529\pi\)
−0.951247 + 0.308431i \(0.900196\pi\)
\(24\) −4.73205 + 1.26795i −0.965926 + 0.258819i
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) −2.00000 − 2.00000i −0.392232 − 0.392232i
\(27\) −5.19615 −1.00000
\(28\) − 8.00000i − 1.51186i
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 4.73205 − 1.26795i 0.863950 − 0.231495i
\(31\) 2.00000 + 3.46410i 0.359211 + 0.622171i 0.987829 − 0.155543i \(-0.0497126\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 5.46410 + 1.46410i 0.965926 + 0.258819i
\(33\) −4.50000 + 2.59808i −0.783349 + 0.452267i
\(34\) 1.83013 + 6.83013i 0.313864 + 1.17136i
\(35\) 8.00000i 1.35225i
\(36\) 5.19615 + 3.00000i 0.866025 + 0.500000i
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 1.36603 − 0.366025i 0.221599 − 0.0593772i
\(39\) 3.46410i 0.554700i
\(40\) −5.46410 − 1.46410i −0.863950 − 0.231495i
\(41\) 2.50000 + 4.33013i 0.390434 + 0.676252i 0.992507 − 0.122189i \(-0.0389915\pi\)
−0.602072 + 0.798441i \(0.705658\pi\)
\(42\) −6.92820 + 6.92820i −1.06904 + 1.06904i
\(43\) 9.52628 + 5.50000i 1.45274 + 0.838742i 0.998636 − 0.0522047i \(-0.0166248\pi\)
0.454108 + 0.890947i \(0.349958\pi\)
\(44\) 6.00000 0.904534
\(45\) −5.19615 − 3.00000i −0.774597 − 0.447214i
\(46\) 2.00000 − 2.00000i 0.294884 − 0.294884i
\(47\) 3.00000 − 5.19615i 0.437595 − 0.757937i −0.559908 − 0.828554i \(-0.689164\pi\)
0.997503 + 0.0706177i \(0.0224970\pi\)
\(48\) −3.46410 − 6.00000i −0.500000 − 0.866025i
\(49\) −4.50000 − 7.79423i −0.642857 − 1.11346i
\(50\) −1.36603 − 0.366025i −0.193185 − 0.0517638i
\(51\) 4.33013 − 7.50000i 0.606339 − 1.05021i
\(52\) 2.00000 − 3.46410i 0.277350 − 0.480384i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −1.90192 − 7.09808i −0.258819 − 0.965926i
\(55\) −6.00000 −0.809040
\(56\) 10.9282 − 2.92820i 1.46034 − 0.391298i
\(57\) −1.50000 − 0.866025i −0.198680 − 0.114708i
\(58\) 0 0
\(59\) 0.866025 − 0.500000i 0.112747 − 0.0650945i −0.442566 − 0.896736i \(-0.645932\pi\)
0.555313 + 0.831641i \(0.312598\pi\)
\(60\) 3.46410 + 6.00000i 0.447214 + 0.774597i
\(61\) −10.3923 − 6.00000i −1.33060 − 0.768221i −0.345207 − 0.938527i \(-0.612191\pi\)
−0.985391 + 0.170305i \(0.945525\pi\)
\(62\) −4.00000 + 4.00000i −0.508001 + 0.508001i
\(63\) 12.0000 1.51186
\(64\) 8.00000i 1.00000i
\(65\) −2.00000 + 3.46410i −0.248069 + 0.429669i
\(66\) −5.19615 − 5.19615i −0.639602 − 0.639602i
\(67\) −2.59808 + 1.50000i −0.317406 + 0.183254i −0.650236 − 0.759733i \(-0.725330\pi\)
0.332830 + 0.942987i \(0.391996\pi\)
\(68\) −8.66025 + 5.00000i −1.05021 + 0.606339i
\(69\) −3.46410 −0.417029
\(70\) −10.9282 + 2.92820i −1.30617 + 0.349987i
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −2.19615 + 8.19615i −0.258819 + 0.965926i
\(73\) 9.00000 1.05337 0.526685 − 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 2.73205 − 0.732051i 0.317594 − 0.0850992i
\(75\) 0.866025 + 1.50000i 0.100000 + 0.173205i
\(76\) 1.00000 + 1.73205i 0.114708 + 0.198680i
\(77\) 10.3923 − 6.00000i 1.18431 − 0.683763i
\(78\) −4.73205 + 1.26795i −0.535799 + 0.143567i
\(79\) 7.00000 − 12.1244i 0.787562 − 1.36410i −0.139895 − 0.990166i \(-0.544677\pi\)
0.927457 − 0.373930i \(-0.121990\pi\)
\(80\) − 8.00000i − 0.894427i
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) −5.00000 + 5.00000i −0.552158 + 0.552158i
\(83\) −3.46410 − 2.00000i −0.380235 − 0.219529i 0.297686 − 0.954664i \(-0.403785\pi\)
−0.677920 + 0.735135i \(0.737119\pi\)
\(84\) −12.0000 − 6.92820i −1.30931 − 0.755929i
\(85\) 8.66025 − 5.00000i 0.939336 − 0.542326i
\(86\) −4.02628 + 15.0263i −0.434165 + 1.62033i
\(87\) 0 0
\(88\) 2.19615 + 8.19615i 0.234111 + 0.873713i
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 2.19615 − 8.19615i 0.231495 − 0.863950i
\(91\) − 8.00000i − 0.838628i
\(92\) 3.46410 + 2.00000i 0.361158 + 0.208514i
\(93\) 6.92820 0.718421
\(94\) 8.19615 + 2.19615i 0.845369 + 0.226516i
\(95\) −1.00000 − 1.73205i −0.102598 − 0.177705i
\(96\) 6.92820 − 6.92820i 0.707107 − 0.707107i
\(97\) −0.500000 + 0.866025i −0.0507673 + 0.0879316i −0.890292 − 0.455389i \(-0.849500\pi\)
0.839525 + 0.543321i \(0.182833\pi\)
\(98\) 9.00000 − 9.00000i 0.909137 − 0.909137i
\(99\) 9.00000i 0.904534i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 72.2.n.a.61.2 yes 4
3.2 odd 2 216.2.n.a.181.1 4
4.3 odd 2 288.2.r.a.241.1 4
8.3 odd 2 288.2.r.a.241.2 4
8.5 even 2 inner 72.2.n.a.61.1 yes 4
9.2 odd 6 648.2.d.a.325.2 2
9.4 even 3 inner 72.2.n.a.13.1 ✓ 4
9.5 odd 6 216.2.n.a.37.2 4
9.7 even 3 648.2.d.d.325.1 2
12.11 even 2 864.2.r.a.721.1 4
24.5 odd 2 216.2.n.a.181.2 4
24.11 even 2 864.2.r.a.721.2 4
36.7 odd 6 2592.2.d.b.1297.2 2
36.11 even 6 2592.2.d.a.1297.1 2
36.23 even 6 864.2.r.a.145.2 4
36.31 odd 6 288.2.r.a.49.2 4
72.5 odd 6 216.2.n.a.37.1 4
72.11 even 6 2592.2.d.a.1297.2 2
72.13 even 6 inner 72.2.n.a.13.2 yes 4
72.29 odd 6 648.2.d.a.325.1 2
72.43 odd 6 2592.2.d.b.1297.1 2
72.59 even 6 864.2.r.a.145.1 4
72.61 even 6 648.2.d.d.325.2 2
72.67 odd 6 288.2.r.a.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.n.a.13.1 ✓ 4 9.4 even 3 inner
72.2.n.a.13.2 yes 4 72.13 even 6 inner
72.2.n.a.61.1 yes 4 8.5 even 2 inner
72.2.n.a.61.2 yes 4 1.1 even 1 trivial
216.2.n.a.37.1 4 72.5 odd 6
216.2.n.a.37.2 4 9.5 odd 6
216.2.n.a.181.1 4 3.2 odd 2
216.2.n.a.181.2 4 24.5 odd 2
288.2.r.a.49.1 4 72.67 odd 6
288.2.r.a.49.2 4 36.31 odd 6
288.2.r.a.241.1 4 4.3 odd 2
288.2.r.a.241.2 4 8.3 odd 2
648.2.d.a.325.1 2 72.29 odd 6
648.2.d.a.325.2 2 9.2 odd 6
648.2.d.d.325.1 2 9.7 even 3
648.2.d.d.325.2 2 72.61 even 6
864.2.r.a.145.1 4 72.59 even 6
864.2.r.a.145.2 4 36.23 even 6
864.2.r.a.721.1 4 12.11 even 2
864.2.r.a.721.2 4 24.11 even 2
2592.2.d.a.1297.1 2 36.11 even 6
2592.2.d.a.1297.2 2 72.11 even 6
2592.2.d.b.1297.1 2 72.43 odd 6
2592.2.d.b.1297.2 2 36.7 odd 6